On Spectral Estimates for Schrodinger-Type Operators: The Case of Small Local Dimension

The behavior of the discrete spectrum of the Schrodinger operator -Delta - V is determined to a large extent by the behavior of the corresponding heat kernel P(t; x, y) as t -> 0 and t -> infinity. If this behavior is power-like, i.e., parallel to P(t; .,.)parallel to(L)infinity = O(t(-delta/2)), t -> 0, parallel to P(t; .,,)parallel to(L)infinity = O(t-(D/2)), t -> infinity, then it is natural to call the exponents delta and D the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where delta < D, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.